% \begin{figure*}[t!]
%   \begin{center}
%     \subfigure[Revenue of the leader and the follower]{
%       \includegraphics[width=0.5\textwidth]{fig/duorev.eps}
%       \label{Flo:duorev}
%     }\\
%     \subfigure[Market share and subscription share of the leader]{
%       \includegraphics[width=0.5\textwidth]{fig/duoshare.eps}
%        \label{Flo:duoshare}
%    }\\
%     \subfigure[Prices of the leader and follower]{
%       \includegraphics[width=0.5\textwidth]{fig/duopoly_p.eps}
%        \label{Flo:duoprice}
%    }
%   \end{center}
%   \caption{Revenue, market share, and prices for the duopoly cases}
%   \label{Flo:duopoly}
% \end{figure*}


\section{Duopoly}  \label{sec:duopoly}

This section is devoted to understanding how the access market is shared
between two operators when the second one enters the market posterior to
the first one. This is an interesting and practical issue because in
reality mobile network operators' main concern on the femtocell services
may be the decision of when they start to offer the service.

\subsection{Game Model}
We assume that both operators offer only one service with
flat pricing for the sake of simplicity. % Considering the results of
% previous section, these assumptions are not too restrictive.
We assume that the incumbent offers \emph{open-femto} service with flat
price $p_1$. For the entrant, we consider two cases: the entrant decides
to offer either \emph{open-femto} or \emph{mobile-only} service with
flat price of $p_2$. We model this case with a four-stage sequential
game. 
\begin{separation}
\begin{compactenum}[\bf \em {Stage} 1.]
\item   The leader (incumbent) decides on a flat price $p_1$ for the
\emph{open-femto} service.

\item The users make decision to join the service or not for the 
  price $p_1$ selected by the leader.

\item Given $p_1,$ the entrant decides on a flat price $p_2$ for its
  selected service.  

% its service,{\em
%   open-femto} or {\em mobile-only}, and a flat price $p_2$ for the
% decided service. 

\item The users finally make new decisions for their subscription, given
  $p_1$ and $p_2.$ The users of the incumbent decides whether to
  switch the provider and/or the service or not. The users outside the
  market select one of two providers and its service, or just stay with no
  subscription.
\end{compactenum}
\end{separation}

% \smallskip
% \noindent{\bf \em Stage 1:} The leader (incumbent) decides a flat price $p_1$ for the
% \emph{open-femto} service.

% \smallskip
% \noindent{\bf \em Stage 2:} The users make decisions to join the service
% or not for the given price $p_1$ by the leader. 

% \smallskip
% \noindent{\bf \em Stage 3:} The entrant decides its service out of {\em
%   open-femto} and {\em mobile-only} and the flat price $p_2$ for the
% decided service. 

% \smallskip
% \noindent{\bf \em Stage 4:} The users finally make new decisions to join
% service or to switch a provider given $p^1$ and $p^2$.  The users of the
% incumbent decides whether or not to switch the service. The other users
% either select one of the two service or stay unsubscribed.

% The game is a sequential one consisting of three stages. In the first
% stage,the leader (incumbent) decides the flat price $p^1$ for the
% \emph{open-femto} service, in the second stage, the entrant decides its
% service and service charge $p^2$. In the third stage, users make new
% decisions to join service or to switch a provider given $p^1$ and $p^2$.
% The users of the incumbent decides whether or not to switch the
% service. The other users either select one of the two service or stay
% unsubscribed.

The leader first selects the price $p_1$ expecting competition
with the follower, where $p_1$ usually is lower than the price in the
monopoly. Given $p_1$, the entrant decides on $p_2$ which is expected be
smaller than $p_1.$ This is due to the {\em conjecture} that the entrant's service has worse quality than the
leader's  one due to less number of femto BSs, but he/she wants to attract
the leader's users or even new users (We later present that the numerical
result contradicts to this conjecture at equilibrium).
However, considering positive
externality of the \emph{open-femto} users, the subscription ratio of
the leader plays favorably to the incumbent\footnote{leader=incumbent.}. As the service charge of
the new entrant is lower, some users will select the entrant, which will
increase the capacity of the entrant's network.

% We repeat the selection of the
% user choice until user selection process stabilizes, i.e., until no
% users changes its subscription decision.

\subsection{Numerical Results}

\begin{figure}[t!]
   \begin{center}
       \includegraphics[width=0.7\columnwidth]{fig/duorev.eps}
  \end{center}
       \caption{Revenue of the leader and the follower}
       \label{Flo:duorev}
\end{figure}

\begin{figure}[t!]
      \begin{center}
       \includegraphics[width=0.7\columnwidth]{fig/duoshare.eps}
\end{center}
       \caption{Market share and subscription share of the leader}
       \label{Flo:duoshare}
\end{figure}

\begin{figure}
	\begin{center}
       \includegraphics[width=0.7\columnwidth]{fig/duopoly_p.eps}
	\end{center}
       \caption{Prices of the leader and follower}
       \label{Flo:duoprice}
\end{figure}



It is almost impossible to analytically solve this game
and compute the equilibrium due to complex coupling of users' choices
and pricing strategies of two providers: Users' choices affect the
capacity of each operator, which in turn impacts choices of other users.
We rely on numerical approaches to find the equilibrium by solving a
multiple of fixed point problems.  

Figs.~\ref{Flo:duorev} and \ref{Flo:duoshare} show the providers'
revenue and the market shares, where `O-O' means that both offer
\emph{open-femto} services and `O-M' means that the entrant provides
only macro BSs.  As we see, when both offer the \emph{open-femto}
services, the revenue and the market share of the incumbent surpass
those of the entrant, since the leader initially gathers a larger share
of market with the large system capacity according to the open femto BSs
(again, positive externality).  Fig.~\ref{Flo:duoprice} shows that the incumbent tries to
secure as many users as possible with a low price, and from
Fig.~\ref{Flo:duoshare}, most users select the leader in `O-O'
game. Thus, operators should introduce the \emph{open-femto} service as
soon as possible to dominate the market. However, when the femto cost exceeds 0.3, the
entrant makes more revenue than the incumbent. The main reason of this
situation is that the entrant could not be dominant provider of the market  as the femto cost increases, since the incumbent should set price to be higher than the
femto cost.

 An interesting result is that for most femto costs the
follower chooses the price $p_2,$ {\em higher} than $p_1$ equilibrium, see
Fig.~\ref{Flo:duoprice}. This phenomenon is caused by the follower's
economic choice of {\em avoiding competition,} because the leader's network
has huge capacity (due to many open-femto users) and the follower tries
to attain its economic benefit just by selecting a {\em high} price and
attract the users with large $\gamma.$  

Observe that the entrant is better off by introducing
\emph{mobile-only} service rather than \emph{open-femto} service.  
This is somewhat surprising because
{\em open-femto} was always good in the monopoly
%.The entrant makes even more revenue than the
%incumbent when the femto cost exceeds 0.3 
%because of the following two reasons. 
There are two reasons for the observations.
First, since the incumbent provides the service
with higher prices, the entrant attracts users of low $\gamma$ type
with low prices. As Fig.~\ref{Flo:duoprice} shows, the price of the
incumbent is regarded as the price for monopoly, since the capacity of the entrant without
femto BSs is much less than that of the incumbent. Second, since the entrant
should set the price to be higher than the femto cost, the entrant
becomes more attractive as the femto cost increases. Note that the big
difference in the total revenue between `O-O' and `O-M' in
Fig.~\ref{Flo:duorev} is due to the impact of competition. 

% \begin{figure}[t!]
%       \begin{center}
%        \includegraphics[width=0.9\columnwidth]{fig/coduo_rev.eps}
% \end{center}
%        \caption{Revenue of duopoly market with coalition}
%        \label{Flo:duopolyco}
% \end{figure}

% \begin{figure}
% 	\begin{center}
%        \includegraphics[width=0.9\columnwidth]{fig/coduo_sub.eps}
% 	\end{center}
%        \caption{Subscription ratio of duopoly market with coalition}
%        \label{Flo:duopolycosub}
% \end{figure}

% \textbf{Here is added}

% Now, we will see another scenario for duopoly. According to the above
% results, operator get big advantage when they begin the open femto
% service earlier than the competitor. Thus, we assume
% that two providers have symmetric policy which means that, when one of
% the providers unveils {\em open-femto} service plan, the other provider
% follows with the exactly same policy. This scenario is operated as
% followings. Initially, each provider serves half of subscribing users
% without femto BSs. And then,
% providers introduce {\em open-femto} service at same time with
% same price. Thus, this could be considered as coalition game. As
% Fig.~\ref{Flo:duopolyco} shows, with {\em open-femto} service,
% providers have more revenue. Moreover, as the initial subscription ratio is smaller, they
% could get more benefit since the price for {\em mobile-only} is more expensive. Fig.~\ref{Flo:duopolycosub} shows the subscription
% ratio of each service type. One of interesting results is that the
% subscription ratio increase with {\em open-femto} service, and the
% increament becomes larger as the subscription ratio is smaller. The
% reason is that the open femto BSs give more utility improvement to {/em
% mobile-only} users, since per user throughput of open femto BSs is decreasing
% function of the number of users. 




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